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\usepackage{amsmath, amssymb} % 数学公式与符号
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\title{复变函数第1章：复数与复变函数}
\author{ZYQ ET AL}
\date{2024年3月9日}

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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

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\begin{frame}{第1章目录 }


\begin{enumerate}

\item[1.1.] 复数
\item[1.2.] 复平面上的点集
\item[1.3.] 复变函数
\item[1.4.] 复球面和无穷远点

\end{enumerate}

\end{frame}

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\begin{frame}{例子1.1.  }


\begin{itemize}

\item  {\color{red}问题：将下列复数表示成 $x+iy$ 的形式：
(1) $\left(\frac{1-i}{1+i}\right)^7$. 
(2) $\frac{i}{1-i}+\frac{1-i}{i}$. 
 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.3.  }


\begin{itemize}

\item  {\color{red}问题：求复数的幅角 $Arg(2-2i)$ 以及 $Arg(-3+4i)$. }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.4.  }


\begin{itemize}

\item  {\color{red}问题：已知流体在某点的速度为 $v=-1-i$, 求其大小和方向。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.5.  }


\begin{itemize}

\item  {\color{red}问题：将下述复数化为指数形式：
$1+i, i, 1, -2, -3i, -i, -1, 1$.  
 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.6.  }


\begin{itemize}

\item  {\color{red}问题：将复数 $1-\cos\varphi + i\sin\varphi$ 化为指数形式。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.7.  }


\begin{itemize}

\item  {\color{red}问题：对于复数 $\alpha,\beta$, 证明：若 $\alpha\beta=0$, 则有 $\alpha=0$ 或 $\beta=0$.  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.8.  }


\begin{itemize}

\item  {\color{red}问题：求 $\cos 3\theta$ 与 $\sin 3\theta$ 用 $\cos\theta$ 与 $\sin\theta$ 表示的式子。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.9.  }


\begin{itemize}

\item  {\color{red}问题：计算 $\sqrt[3]{-8}$. }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.10.  }


\begin{itemize}

\item  {\color{red}问题：解方程 $(1+z)^5=(1-z)^5$.  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.11.  }


\begin{itemize}

\item  {\color{red}问题：设复数 $z=x+iy$, 求复数 $w=\frac{1+z}{1-z}$ 的实部、虚部和模。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.12.  }


\begin{itemize}

\item  {\color{red}问题：设 $z_1,z_2$ 是两个复数，证明 
$$|z_1+z_2|^2 = |z_1|^2 + |z_2|^2 + 2Re(z_1\overline{z_2}). $$ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.13.  }


\begin{itemize}

\item  {\color{red}问题：设 $|a|<1$, $|b|<1$, 证明： $$\left\vert \frac{a-b}{1-\bar{a}b} \right\vert <1. $$ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.14.  }


\begin{itemize}

\item  {\color{red}问题：使用复数写出连接两点 $z_1,z_2$ 的线段的参数方程。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.15.  }


\begin{itemize}

\item  {\color{red}问题：使用复数写出以 $z_0$ 为圆心，以 $R$ 为半径的圆周的参数方程。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.16.  }


\begin{itemize}

\item  {\color{red}问题：证明三个复数 $z_1,z_2,z_3$ 成为一个等边三角形的顶点的充分必要条件是
$z_1^2+z_2^2+z_3^2 = z_1z_2 + z_1z_3 + z_2z_3. $ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.17.   }


\begin{itemize}

\item  {\color{red}问题：使用复数证明三角形的内角和为 $\pi$.  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.7. 由实数构造复数的方法的推广  }


\begin{itemize}

\item  {\color{red}问题：什么是四元数系？什么是八元数系？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.18.   }


\begin{itemize}

\item  {\color{red}问题：证明点集 $E$ 的边界 $\partial E$ 是闭集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.19.   }


\begin{itemize}

\item  {\color{red}问题：画出复数表示的平面区域：}
\begin{enumerate}
\item  {\color{red}$|z|<1$.  }
\item  {\color{red}$|z|\le 1$. }
\item  {\color{red}$Im(z)>0$.} 
\item  {\color{red}$y_1<Im(z)<y_2$.}
\item  {\color{red}$r<|z|<R$. }
\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{定义1.7-1.9.  }


\begin{itemize}

\item  {\color{red}问题：} 
\begin{enumerate}
\item  {\color{red}什么是复数平面上的一条连续曲线？}
\item  {\color{red}什么是简单闭曲线？}
\item  {\color{red}什么是可求长的曲线？}
\item  {\color{red}什么是可求长的曲线的长度？}
\item  {\color{red}什么是光滑闭曲线？}
\item  {\color{red}什么是分段光滑曲线？ }
\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.24. }


\begin{itemize}

\item  {\color{red}问题：设简单曲线的参数方程为
$$\left\{
\begin{array}{ll}
x=x(t)=t, \\
y=y(t)=t\sin\frac{1}{t}, \\
\end{array}\right. 0\le t\le 1,
$$
验证这条曲线是不可求长的。
}

%\item  解答：


\end{itemize}

\end{frame}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{定理1.1. 若尔当定理 }


\begin{itemize}

\item  {\color{red}问题：任意简单闭曲线 $C$ 将复数平面唯一地分成 $C$, $I(C)$ 以及 $E(C)$ 三个点集，满足下述性质：
这三个点集互不相交，$I(C)$ 是一个有界区域，$E(C)$ 是一个无界区域，若简单折线 $P$ 的一个端点属于 $I(C)$, 另一个端点属于 $E(C)$, 则 $P$ 必与 $C$ 有交点。
 }

%\item  证明：


\end{itemize}

\end{frame}

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\begin{frame}{定义1.12 }


\begin{itemize}

\item  {\color{red}问题：什么是单值函数？什么是多值函数？  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.25. }


\begin{itemize}

\item  {\color{red}问题：举例说明单值函数与多值函数。  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.26. }


\begin{itemize}

\item  {\color{red}问题：将复变函数 $w=z^2+2$ 写成代数形式和三角形式。  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.27. }


\begin{itemize}

\item  {\color{red}问题：考察函数 $w=\bar{z}$ 所构成的映射。  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.28. }


\begin{itemize}

\item  {\color{red}问题：设有函数 $w=z^2$, 它将 $z$ 平面上的下述曲线变成 $w$ 平面上的何种曲线？  }
\begin{enumerate}
\item  {\color{red} 以原点为圆心，2为半径的在第一象限的圆弧。} 
\item  {\color{red} 倾角为 $\theta=\pi/3$ 的直线。} 
\item  {\color{red} 双曲线 $x^2-y^2=4$. } 
\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{定义1.16. }


\begin{itemize}

\item  {\color{red}问题：什么是复变函数 $w=f(z)$ 当 $z\to z_0$ 时的极限？  }

%\item  解答：


\end{itemize}

\end{frame}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{定理1.2. }


\begin{itemize}

\item  {\color{red}问题：记 $z=x+iy$, 写出复变函数 $f(z) = u(x,y) + iv(x,y)$ 在 $z\to z_0$ 时的极限为 $a+ib$ 的充分必要条件。  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{定义1.17. }


\begin{itemize}

\item  {\color{red}问题：什么时候称复变函数 $w=f(z)$ 在 $z_0$ 连续？  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{定理1.3. }


\begin{itemize}

\item  {\color{red}问题：写出复变函数 $f(z) = u(x,y) + iv(x,y)$ 在 $z_0=x_0+iy_0$ 连续的充分必要条件。   }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.29. }


\begin{itemize}

\item  {\color{red}问题：证明函数 $f(z) = \frac{z}{\bar{z}}$ 当 $z\to 0$ 时的极限不存在。  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.30. }


\begin{itemize}

\item  {\color{red}问题：证明函数 $$f(z) = \frac{1}{2i} \left( \frac{z}{\bar{z}} - \frac{\bar{z}}{z} \right)$$  
在 $z\to 0$ 时的极限不存在。
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{定义1.18. }


\begin{itemize}

\item  {\color{red}问题：什么时候称复变函数 $w=f(z)$ 是连续函数？  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.31. }


\begin{itemize}

\item  {\color{red}问题：设 $\lim\limits_{z\to z_0}f(z)=\eta$, 证明函数 $f(z)$ 在 $z_0$ 的某个去心邻域内是有界的。  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{例子1.32. }


\begin{itemize}

\item  {\color{red}问题：设函数 $f(z)$ 在点 $z_0$ 连续，且 $f(z_0)\neq 0$. 证明 $f(z)$ 在 $z_0$ 的某个邻域内恒不为零。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{定理1.7. }


\begin{itemize}

\item  {\color{red}问题：考虑定义在有界闭集 $E=\{z:|z|\le 1\}$ 上的复变函数 $f(z)=\frac{1}{2-z}$. }

\begin{enumerate}
\item  {\color{red}验证 $f(z)$ 在 $E$ 上是连续的。}
\item  {\color{red}求 $|f(z)|$ 在 $E$ 上的最大值与最小值。}
\item  {\color{red}验证 $f(z)$ 在 $E$ 上是一致连续的。}
\end{enumerate}

%\item  解答：

\end{itemize}

\end{frame}

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\begin{frame}{1.4. 复球面 }


\begin{itemize}

\item  {\color{red}问题：什么是复球面？  }

%\item  解答：


\end{itemize}

\end{frame}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{例子1.33. }


\begin{itemize}

\item  {\color{red}问题：证明函数 
$$f(z)=\left\{
\begin{array}{ll}
\frac{1}{z}, & z\neq 0, \\
\infty, & z=0, \\
0, & z=\infty,
\end{array}\right.
$$  
在扩充复平面上是广义连续的。
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题一1. }


\begin{itemize}

\item  {\color{red}问题：设 $z=\frac{1-\sqrt{3}i}{2}$, 求 $|z|$ 及 $Arg(z)$. %并用指数形式表示 $z$. 
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题一3. }


\begin{itemize}

\item  {\color{red}问题：求方程 $z^3+z^2+z+1=0$ 的根，并将 $z^3+z^2+z+1$ 因式分解。
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题一5. }


\begin{itemize}

\item  {\color{red}问题：设三个复数 $z_1,z_2,z_3$ 符合条件 $z_1+z_2+z_3=0$ 以及 $|z_1|=|z_2|=|z_3|=1$. 
证明这三点是一个内接于单位圆周的正三角形的顶点。
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题一7. }


\begin{itemize}

\item  {\color{red}问题：设 $n>1$ 为正整数，证明 $$\prod_{k=1}^{n-1} \left(x^2-2x\cos\frac{k\pi}{n}+1 \right) = 
\frac{x^{2n}-1}{x^2-1}. $$
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题一10. }


\begin{itemize}

\item  {\color{red}问题：求下述参数方程给出的曲线：
\begin{enumerate}
\item  $z=(1+i)t$. 
\item  $z=a\cos(t)+ib\sin(t)$. 
\item  $z=t+\frac{i}{t}$. 
\item  $z=t^2+\frac{i}{t^2}$. 
\end{enumerate} 


}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题一11. }


\begin{itemize}

\item  {\color{red}问题：函数 $w=\frac{1}{z}$ 将 $z$ 平面上的下述曲线变成 $w$ 平面上的什么曲线？
\begin{enumerate}
\item  $x^2+y^2=4$. 
\item  $y=x$. 
\item  $x=1$. 
\item  $(x-1)^+y^2=1$. 
\end{enumerate} 
 
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题一14. }


\begin{itemize}

\item  {\color{red}问题：设 $f(z)=e^x(\cos y + i\sin y)$, 其中 $z=x+iy$, 问当 $z\to\infty$ 时，$f(z)$ 有无极限，包括广义极限？
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题一15. }


\begin{itemize}

\item  {\color{red}问题：函数 $f(z)=\frac{1}{1-z}$ 在单位圆 $|z|<1$ 内是否连续？是否一致连续？
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题二6. }


\begin{itemize}

\item  {\color{red}问题：设 $|z|=1$, 证明 $$\left\vert \frac{az+b}{\bar{b}z+\bar{a}} \right\vert =1. $$
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题二7. }


\begin{itemize}

\item  {\color{red}问题：已知正方形的两个相对的顶点坐标为 $z_1(0,-1)$ 和 $z_3(2,5)$, 求另两个顶点的坐标。
}

%\item  解答：


\end{itemize}

\end{frame}



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\end{document}



